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Chapter 5: Problem 37

Auro FiNANCING Dan is contemplating trading in his car for a new one. He can afford a monthly payment of at most \(\$ 400 .\) If the prevailing interest rate is \(7.2 \% /\) year compounded monthly for a 48 -mo loan, what is the most expensive car that Dan can afford, assuming that he will receive \(\$ 8000\) for the trade-in?

Short Answer

Expert verified
Dan can afford a new car costing up to \$23,516.40.

Step by step solution

01

Write down given information from the problem

- Maximum Monthly Payment (PMT) = $400 - Interest Rate (i) = 7.2% per year (compounded monthly) - Loan Term = 48 months - Trade-in Value = $8000
02

Convert annual interest rate to monthly interest rate

To find the monthly interest rate (r), we divide the annual interest rate by 12 \( (i = 7.2 \% / 1 \text{ year}) \): \[ r = \frac{i}{12} = \frac{7.2 \%}{12} = 0.6 \% \]
03

Calculate the present value factor using the formula

The present value factor is used to simplify the calculation of the loan amount. The formula for calculating the present value factor (PVF) is: \[ PVF = \frac{1 - (1 + r)^{-n}}{r} \] Where n is the number of months (loan term) Plug in the values of r and n: \[ PVF = \frac{1 - (1 + 0.006)^{-48}}{0.006} \]
04

Calculate the present value factor (PVF)

Now calculate the present value factor: \[ PVF = \frac{1 - (1 + 0.006)^{-48}}{0.006} \approx 38.791 \]
05

Calculate the Loan Amount

Now, we can use the maximum monthly payment (PMT) and the present value factor (PVF) to calculate the loan amount (LA): \[ LA = PMT \times PVF = 400 \times 38.791 = \$ 15516.4 \]
06

Calculate the maximum price Dan can afford for a new car

Finally, add the loan amount (LA) to the trade-in value to find the maximum price Dan can afford for a new car: \[ Maximum\_Price = LA + Trade\_in\_Value = 15516.4 + 8000 = \$ 23516.4 \] Dan can afford a new car costing up to \$23,516.40.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amortization
Amortization in financial mathematics refers to the process of spreading out a loan into a series of fixed payments over time. This concept is crucial when considering car loans, mortgages, or any installment payments. When a borrower like Dan decides on a loan, the amortization schedule will help assure equal payments until the maturity of the loan.
- Each payment installment covers: - A portion of the original loan (principal) - Accumulated interest on the remaining balance The beauty of amortization is the simplicity it provides to the borrower; they know exactly what to pay each period. This clarity helps in budgeting and financial planning, ensuring that Dan can confidently afford his monthly car payments.
Present Value Factor
The present value factor (PVF) is a key concept in loan calculations. It helps convert a series of future cash flows into their present value. For a loan, it simplifies determining how much can be borrowed given a fixed repayment plan.In Dan's case, the PVF is calculated using:- The monthly interest rate- The total number of payments (months)- The formula: \[ PVF = \frac{1 - (1 + r)^{-n}}{r} \]Where \( r \) represents the monthly interest rate and \( n \) is the total number of payments. By understanding the PVF, we can better grasp how future loan payments aggregate into today's loan value.
Monthly Interest Rate
The monthly interest rate is derived from the annual interest rate, crucial for determining Dan鈥檚 loan parameters. Since interest compounding affects monthly payments, we first convert the annual interest rate to a monthly figure. Here's how it's done: - Take the annual interest rate (7.2% in Dan's case) and divide by 12. - This results in a monthly interest rate of 0.6%. The monthly interest affects both the amount Dan pays monthly and the loan balance over time. Frequent compounding means even small changes can significantly impact the total loan cost. This is why understanding and accurately applying the monthly interest rate is so critical.
Loan Calculation
Calculating the loan amount requires combining several financial principles, including the present value factor and the maximum affordable monthly payment.Here鈥檚 a step-by-step to arrive at Dan鈥檚 situation:- **Monthly Payment**: Given as \(400.- **Present Value Factor**: Already calculated as 38.791.- **Loan Amount**: Multiply the monthly payment by PVF: \[ LA = 400 \times 38.791 = \\) 15516.4 \]Take into account Dan's trade-in value:- **Trade-in**: \\(8,000.- **Maximum Affordable Car Price**: Combine the loan and trade-in values: \[ Maximum\_Price = 15516.4 + 8000 = \\) 23516.4 \]This comprehensive calculation ensures Dan understands both his borrowing capacity and realistic budget for purchasing a new vehicle.

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Most popular questions from this chapter

FiNANCING A HomE The Taylors have purchased a \(\$ 270,000\) house. They made an initial down payment of \(\$ 30,000\) and secured a mortgage with interest charged at the rate of \(8 \% / y\) ear on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over \(30 \mathrm{yr}\), what monthly payment will the Taylors be required to make? What is their equity (disregarding appreciation) after 5 yr? After 10 yr? After 20 yr?

SINKING FuNDS Lowell Corporation wishes to establish a sinking fund to retire a \(\$ 200,000\) debt that is due in \(10 \mathrm{yr}\). If the investment will earn interest at the rate of \(9 \% /\) year compounded quarterly, find the amount of the quarterly deposit that must be made in order to accumulate the required sum.

A culture of a certain bacteria is known to double in number every \(3 \mathrm{hr}\). If the culture has an initial count of 20 , what will be the population of the culture at the end of \(24 \mathrm{hr}\) ?

Jodie invested \(\$ 15,000\) in a mutual fund 4 yr ago. If the fund grew at the rate of \(9.8 \%\) lyear compounded monthly, what would Jodie's account be worth today?

STUDENT LoANS Joe secured a loan of \(\$ 12,0003\) yr ago from a bank for use toward his college expenses. The bank charged interest at the rate of \(4 \% /\) year compounded monthly on his loan. Now that he has graduated from college, Joe wishes to repay the loan by amortizing it through monthly payments over \(10 \mathrm{yr}\) at the same interest rate. Find the size of the monthly payments he will be required to make.
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